Probability theory Poisson distribution

A stochastic theory provides a simple model for chromatography.11 The term stochastic implies the presence of a random variable. The model supposes that, as a molecule travels through a column, it spends an average time Tm in the mobile phase between adsorption events. The time between desorption and the next adsorption is random, but the average time is Tm. The average time spent adsorbed to the stationary phase between one adsorption and one desorption is rs. While the molecule is adsorbed on the stationary phase, it does not move. When the molecule is in the mobile phase, it moves with the speed ux of the mobile phase. The probability that an adsorption or desorption occurs in a given time follows the Poisson distribution, which was described briefly in Problem 19-21. 

With the additional assumption of the Poisson distribution for the probability of n jumps in a time t (mean time between jumps r), Equations 8 and 9 can be used in existing lattice diffusion theories to calculate relaxation times (approximately, i.e., within 10%) (16,18). 

The central point of the active-site theory is that the areal density of nanoscale surface features is small and approaches the same scale as individual particle surface areas, especially in the regime of nanoparticles. The probability that a particle bears i active-sites is given by the Poisson distribution, as follows ... 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

Given these assumptions, the distribution of the observed total number of counts according to probability theory should be binomial with parameters N and p. Because p is so small, this binomial distribution is approximated very well by the Poisson distribution with parameter Np, which has a mean of Np, and a standard deviation of Np. The mean and variance of a Poisson distribution are numerically equal so, a single counting measurement provides an estimate of the mean of the distribution Np and its square root is an estimate of the standard deviation /Np. When this Poisson approximation is valid, one may estimate the standard uncertainty of the counting measurement without repeating the measurement (a Type B evaluation of uncertainty). 

Due to influences of measurement error, manufacturing, assembly and other factors, there is uncertainty in mechanical components geometry size, material properties parameters (such as elastic modulus, Poisson s ratio), and so on. The uncertainty significantly affects the reliability of the mechanical components. Therefore, it is important to choose appropriate distributions of random variables before wear reliability analysis. The traditional probability theory method is one of common methods which are used to deal with the uncertainty of variables. However, the method of probability and statistics is subject to restrictions of sample size, sampling time and sample conditions. Sometimes, because sample size is too small, it is impossible to obtain the probability density functions of random variables. 

We first review fundamentals of the theory of stochastic processes. The system dynamics are specified by the set of its states, 5, and the transitions between them, S -> S, where S,S e 5. For example, the state S can denote the position of a Brownian particle, the numbers of molecules of different chemical species, or any other variable that characterizes the state of the system of interest. Here we restrict ourselves to processes for which the transition rates depend only on the system s instantaneous state, andnotontheentirety of its history. Such memoryless processes are known as Markovian and are applicable to a wide range of systems. We also assume that the transition rates do not explicitly depend on time, a condition known as stationarity. In this review we make the standard assumption that the transitions between the states are Poisson distributed random processes. In other words, the probability of transitioning from state S to state 5 in an infinitesimal interval, dt, is a S,S )dt, where a(S,S ) is the transition rate. 

The two can be related using a concept called a Poisson process. From the Wikipedia s article on the Poisson Process In probability theory, a Poisson process is a stochastic process which counts the number of events and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter G the parameter is the occurrence rate per unit time] and each of these inter-arrival times is. .. independent of other... 

Fortunately statistical theory provides the necessary tools. Statisticians usually claim that accident occurrence is explained by the Poisson distribution. The probability that a railroad will have x adverse safety occurrences in a given year is given by the formula ... 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

Evans [2] calculated the expectancy of the Poisson probability distribution for the constant propagation rate of domains and two simple nucleation modes instantaneous and spontaneous with the constant rate, F i) = B. Billon et al. [13] extended this approach to the case of time-dependent nucleation rate. According to the Evans theory, an arbitrarily chosen point A can be reached before time t by growing spheres nucleated around it in a distance r (precisely in a distance within the interval (r, r + dr)) before time t - rIG their number is equal to an integral of the nucleation rate F(t) over the time interval (0, t - r/G), multiplied by the considered volume, Artr dr. The total number of spheres occluding the point A until time t is calculated by second integration, over a distance ... 

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